Question : how do you find Cofactor of Matrix?
Solution :
Cofactor of Matrix
Cofactor of a matrix \(A\) is a matrix \(c_{i,j}\) where \(c_{i,j} =(-1)^{i+j}M_{i,j}\). It is denoted as \(cof(A)\).
Example : Find cofactor of the matrix \[\left( {\begin{array}{*{20}{c}}2&3&-1\\4&0&5\\6&0&-2\end{array}} \right) \]
Solution :
For \(A=\left( {\begin{array}{*{20}{c}}2&3&-1\\4&0&5\\6&0&-2\end{array}} \right)\). it’s minors are
\(M_{1,1} = det\left( S_{1,1}\right) =0,M_{1,2} =det\left( S_{1,2}\right)=-38,M_{1,3} = det\left( S_{1,3}\right)=0,M_{2,1}=det\left( S_{2,1}\right)=-6\)
where submatrices are obtained as \(S_{i,j}\) of order \(n-1\) is a matrix obtained by deleting \(i_{th}\) row and \(j_{th}\) column of \(A\).
\(M_{2,2}=det\left( S_{2,2}\right)=2,M_{2,3}=det\left( S_{2,3}\right)=-18,M_{3,1}=det\left( S_{3,1}\right)=15,M_{3,2}=det\left( S_{3,2}\right)=14,M_{3,3}=det\left( S_{3,3}\right)=-12\)
So the cofactor matrix is
\[cof \left( {\begin{array}{*{20}{c}}2&3&-1\\4&0&5\\6&0&-2\end{array}} \right)=\left( {\begin{array}{*{20}{c}}0&38&0\\6&2&18\\15&-14&-12\end{array}} \right)\]